Partially Nonclassical Method and Conformal Invariance in the Context of the Lie Group Method

Abstract

The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the ‘classical’ method, in which the invariant surface condition is not used, and from the ‘nonclassical’ method, in which all the differential consequences are used. It provides additional possibilities for the symmetry analysis of partial differential equations (PDEs), as compared with the ‘classical’ and ‘nonclassical’ methods, in the so-named no-go case when the group generator, associated with one of the independent variables, is identically zero. The method is applied to the flat steady-state boundary layer problem, reduced to an equation for the stream function, and it is found that applying the partially nonclassical method in the no-go case yields new symmetry reductions and new exact solutions of the boundary layer equations. A computationally convenient unified framework for the classical, nonclassical and partially nonclassical methods (λ-formulation) is developed. The issue of conformal invariance in the context of the Lie group method is considered, stemming from the observation that the classical Lie method procedure yields transformations not leaving the differential polynomial of the PDE invariant but modifying it by a conformal factor. The physical contexts, in which that observation could be important, are discussed using the derivation of the Lorentz transformations of special relativity as an example.